Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 85.8%
Time: 23.0s
Alternatives: 16
Speedup: TODO×

Specification

?
\[\begin{array}{l} t_1 := \ell \cdot \ell\\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(t_1 + 2 \cdot \left(t \cdot t\right)\right) - t_1}} \end{array} \]

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 16 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Alternative 1?

\[\begin{array}{l} t_1 := \sqrt{\frac{x + 1}{x + -1}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-233}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot t\right) + t_2\right)}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-162} \lor \neg \left(t \leq 5.8 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + t_2\right)}}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if t < -8.19999999999999944e61

    1. Initial program 32.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*32.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified32.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in t around -inf 96.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
    5. Step-by-step derivation

      1. mul-1-neg96.4%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in96.4%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

    6. Simplified96.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

    if -8.19999999999999944e61 < t < 1.99999999999999992e-233

    1. Initial program 28.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/28.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified28.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 65.9%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified65.9%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 65.9%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-165.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified65.9%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity65.9%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in65.9%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr65.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity65.9%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. fma-def65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      3. *-commutative65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, \color{blue}{t \cdot \frac{t}{x}}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}} \cdot t \]

      4. distribute-rgt-neg-out65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{\color{blue}{-\ell \cdot \ell}}{x}\right)}} \cdot t \]

      5. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{-\color{blue}{{\ell}^{2}}}{x}\right)}} \cdot t \]

      6. distribute-frac-neg65.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      7. unpow265.9%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

      8. associate-*l/73.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \left(-\color{blue}{\frac{\ell}{x} \cdot \ell}\right)\right)}} \cdot t \]

      9. distribute-rgt-neg-in73.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \cdot t \]

    13. Simplified73.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}}} \cdot t \]

    14. Taylor expanded in x around inf 73.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \color{blue}{{t}^{2}} - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}} \cdot t \]
    15. Step-by-step derivation

      1. unpow273.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \color{blue}{\left(t \cdot t\right)} - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}} \cdot t \]

    16. Simplified73.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \color{blue}{\left(t \cdot t\right)} - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}} \cdot t \]

    if 1.99999999999999992e-233 < t < 5.00000000000000014e-162 or 5.80000000000000043e124 < t

    1. Initial program 8.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*8.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified8.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in l around 0 92.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
    5. Step-by-step derivation

      1. +-commutative92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

    6. Simplified92.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

    if 5.00000000000000014e-162 < t < 5.80000000000000043e124

    1. Initial program 49.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/49.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified49.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 76.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+76.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow276.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out76.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow276.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow276.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/76.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg76.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow276.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative76.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow276.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef76.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified76.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 76.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/76.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-176.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow276.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified76.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity76.0%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in76.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr76.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity76.0%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. fma-def76.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      3. *-commutative76.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, \color{blue}{t \cdot \frac{t}{x}}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}} \cdot t \]

      4. distribute-rgt-neg-out76.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{\color{blue}{-\ell \cdot \ell}}{x}\right)}} \cdot t \]

      5. unpow276.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{-\color{blue}{{\ell}^{2}}}{x}\right)}} \cdot t \]

      6. distribute-frac-neg76.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      7. unpow276.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

      8. associate-*l/88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \left(-\color{blue}{\frac{\ell}{x} \cdot \ell}\right)\right)}} \cdot t \]

      9. distribute-rgt-neg-in88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \cdot t \]

    13. Simplified88.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}}} \cdot t \]
  3. Recombined 4 regimes into one program.

  4. Final simplification85.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-233}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot t\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-162} \lor \neg \left(t \leq 5.8 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \end{array} \]

Alternative 2?

\[\begin{array}{l} t_1 := \sqrt{\frac{x + 1}{x + -1}}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-231} \lor \neg \left(t \leq 3.5 \cdot 10^{-162}\right) \land t \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot t\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot t_1}\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if t < -8.19999999999999944e61

    1. Initial program 32.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*32.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def32.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified32.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in t around -inf 96.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
    5. Step-by-step derivation

      1. mul-1-neg96.4%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in96.4%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative96.4%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

    6. Simplified96.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

    if -8.19999999999999944e61 < t < 6.19999999999999976e-231 or 3.4999999999999999e-162 < t < 4.5000000000000004e124

    1. Initial program 35.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/36.0%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified36.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 69.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+69.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow269.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out69.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow269.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow269.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/69.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg69.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow269.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative69.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow269.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef69.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified69.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 69.5%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/69.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-169.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow269.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified69.5%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity69.5%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in69.5%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr69.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity69.5%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. fma-def69.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      3. *-commutative69.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, \color{blue}{t \cdot \frac{t}{x}}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}} \cdot t \]

      4. distribute-rgt-neg-out69.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{\color{blue}{-\ell \cdot \ell}}{x}\right)}} \cdot t \]

      5. unpow269.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{-\color{blue}{{\ell}^{2}}}{x}\right)}} \cdot t \]

      6. distribute-frac-neg69.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      7. unpow269.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

      8. associate-*l/79.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \left(-\color{blue}{\frac{\ell}{x} \cdot \ell}\right)\right)}} \cdot t \]

      9. distribute-rgt-neg-in79.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \cdot t \]

    13. Simplified79.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}}} \cdot t \]

    14. Taylor expanded in x around inf 79.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \color{blue}{{t}^{2}} - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}} \cdot t \]
    15. Step-by-step derivation

      1. unpow279.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \color{blue}{\left(t \cdot t\right)} - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}} \cdot t \]

    16. Simplified79.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \color{blue}{\left(t \cdot t\right)} - \frac{\ell}{x} \cdot \left(-\ell\right)\right)}} \cdot t \]

    if 6.19999999999999976e-231 < t < 3.4999999999999999e-162 or 4.5000000000000004e124 < t

    1. Initial program 8.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*8.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def8.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified8.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in l around 0 92.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
    5. Step-by-step derivation

      1. +-commutative92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative92.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

    6. Simplified92.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification85.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-231} \lor \neg \left(t \leq 3.5 \cdot 10^{-162}\right) \land t \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot t\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 3?

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -0.00024:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \ell\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-160} \lor \neg \left(t \leq 1.25 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot t_1}}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if t < -2.40000000000000006e-4

    1. Initial program 34.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/34.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 19.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow242.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified42.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg91.4%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/91.4%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity91.4%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac91.4%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified91.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -2.40000000000000006e-4 < t < 3.59999999999999997e-237

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-166.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. div-inv66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\ell \cdot \ell\right) \cdot \frac{1}{x}}\right)}} \cdot t \]

      2. distribute-rgt-neg-in66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(\ell \cdot \left(-\ell\right)\right)} \cdot \frac{1}{x}\right)}} \cdot t \]

    11. Applied egg-rr66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(\ell \cdot \left(-\ell\right)\right) \cdot \frac{1}{x}}\right)}} \cdot t \]

    if 3.59999999999999997e-237 < t < 2.80000000000000016e-160 or 1.25000000000000005e-11 < t

    1. Initial program 25.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*25.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in l around 0 88.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
    5. Step-by-step derivation

      1. +-commutative88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

    6. Simplified88.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

    if 2.80000000000000016e-160 < t < 1.25000000000000005e-11

    1. Initial program 41.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/41.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified41.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-178.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. associate-+r-78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}}} \cdot t \]

      3. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\color{blue}{\frac{\ell \cdot \ell}{x}} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      4. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{\color{blue}{{\ell}^{2}}}{x} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      5. fma-udef78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      6. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{{t}^{2}} + \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      7. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(\frac{t}{x} \cdot t + {t}^{2}\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      8. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{\frac{t \cdot t}{x}} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      9. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{\color{blue}{{t}^{2}}}{x} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      10. sub-neg78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) + \left(-\frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

    13. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}} \cdot t \]
  3. Recombined 4 regimes into one program.

  4. Final simplification81.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00024:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \ell\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-160} \lor \neg \left(t \leq 1.25 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \end{array} \]

Alternative 4?

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ \mathbf{if}\;t \leq -0.0011:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-235}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \ell\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-160} \lor \neg \left(t \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot t_1}}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if t < -0.00110000000000000007

    1. Initial program 34.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*34.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def34.6%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified34.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in t around -inf 91.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
    5. Step-by-step derivation

      1. mul-1-neg91.5%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in91.5%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative91.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg91.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval91.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative91.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

    6. Simplified91.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

    if -0.00110000000000000007 < t < 1.55e-235

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-166.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. div-inv66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\ell \cdot \ell\right) \cdot \frac{1}{x}}\right)}} \cdot t \]

      2. distribute-rgt-neg-in66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(\ell \cdot \left(-\ell\right)\right)} \cdot \frac{1}{x}\right)}} \cdot t \]

    11. Applied egg-rr66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(\ell \cdot \left(-\ell\right)\right) \cdot \frac{1}{x}}\right)}} \cdot t \]

    if 1.55e-235 < t < 2.80000000000000016e-160 or 5.00000000000000018e-11 < t

    1. Initial program 25.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-/l*25.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def25.4%

        \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

    3. Simplified25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

    4. Taylor expanded in l around 0 88.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
    5. Step-by-step derivation

      1. +-commutative88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative88.3%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

    6. Simplified88.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

    if 2.80000000000000016e-160 < t < 5.00000000000000018e-11

    1. Initial program 41.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/41.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified41.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-178.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. associate-+r-78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}}} \cdot t \]

      3. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\color{blue}{\frac{\ell \cdot \ell}{x}} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      4. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{\color{blue}{{\ell}^{2}}}{x} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      5. fma-udef78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      6. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{{t}^{2}} + \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      7. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(\frac{t}{x} \cdot t + {t}^{2}\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      8. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{\frac{t \cdot t}{x}} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      9. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{\color{blue}{{t}^{2}}}{x} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      10. sub-neg78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) + \left(-\frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

    13. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}} \cdot t \]
  3. Recombined 4 regimes into one program.

  4. Final simplification81.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0011:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-235}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \ell\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-160} \lor \neg \left(t \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \end{array} \]

Alternative 5?

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -0.00086:\\ \;\;\;\;t \cdot \frac{-t_1}{t}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-238}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \ell\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-160} \lor \neg \left(t \leq 2.65 \cdot 10^{-11}\right):\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot t_2}}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if t < -8.59999999999999979e-4

    1. Initial program 34.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/34.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 19.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow242.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified42.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg91.4%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/91.4%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity91.4%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac91.4%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified91.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -8.59999999999999979e-4 < t < 1.18e-238

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-166.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. div-inv66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\ell \cdot \ell\right) \cdot \frac{1}{x}}\right)}} \cdot t \]

      2. distribute-rgt-neg-in66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(\ell \cdot \left(-\ell\right)\right)} \cdot \frac{1}{x}\right)}} \cdot t \]

    11. Applied egg-rr66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(\ell \cdot \left(-\ell\right)\right) \cdot \frac{1}{x}}\right)}} \cdot t \]

    if 1.18e-238 < t < 2.60000000000000003e-160 or 2.6499999999999999e-11 < t

    1. Initial program 25.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 18.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow233.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified33.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 88.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/88.1%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative88.1%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg88.1%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval88.1%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity88.1%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified88.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if 2.60000000000000003e-160 < t < 2.6499999999999999e-11

    1. Initial program 41.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/41.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified41.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-178.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. associate-+r-78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}}} \cdot t \]

      3. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\color{blue}{\frac{\ell \cdot \ell}{x}} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      4. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{\color{blue}{{\ell}^{2}}}{x} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      5. fma-udef78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      6. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{{t}^{2}} + \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      7. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(\frac{t}{x} \cdot t + {t}^{2}\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      8. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{\frac{t \cdot t}{x}} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      9. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{\color{blue}{{t}^{2}}}{x} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      10. sub-neg78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) + \left(-\frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

    13. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}} \cdot t \]
  3. Recombined 4 regimes into one program.

  4. Final simplification80.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00086:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-238}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \ell\right) \cdot \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-160} \lor \neg \left(t \leq 2.65 \cdot 10^{-11}\right):\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \end{array} \]

Alternative 6?

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -0.000135:\\ \;\;\;\;t \cdot \frac{-t_1}{t}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-160} \lor \neg \left(t \leq 5.1 \cdot 10^{-18}\right):\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot t_2}}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if t < -1.35000000000000002e-4

    1. Initial program 34.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/34.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 19.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow242.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified42.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg91.4%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/91.4%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity91.4%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac91.4%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified91.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -1.35000000000000002e-4 < t < 8.49999999999999951e-237

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-166.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]

    10. Taylor expanded in l around 0 66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    11. Step-by-step derivation

      1. unpow266.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - -1 \cdot \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

      2. associate-*l/66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - -1 \cdot \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)}\right)}} \cdot t \]

      3. neg-mul-166.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{\ell}{x} \cdot \ell\right)}\right)}} \cdot t \]

      4. distribute-rgt-neg-in66.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \cdot t \]

    12. Simplified66.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \cdot t \]

    if 8.49999999999999951e-237 < t < 2.99999999999999997e-160 or 5.09999999999999983e-18 < t

    1. Initial program 25.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 18.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow233.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified33.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 88.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/88.1%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative88.1%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg88.1%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval88.1%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity88.1%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified88.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if 2.99999999999999997e-160 < t < 5.09999999999999983e-18

    1. Initial program 41.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/41.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified41.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow279.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef79.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified79.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-178.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in78.7%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity78.7%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. associate-+r-78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}}} \cdot t \]

      3. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\color{blue}{\frac{\ell \cdot \ell}{x}} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      4. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{\color{blue}{{\ell}^{2}}}{x} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      5. fma-udef78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      6. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{{t}^{2}} + \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      7. +-commutative78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(\frac{t}{x} \cdot t + {t}^{2}\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      8. associate-*l/78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{\frac{t \cdot t}{x}} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      9. unpow278.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{\color{blue}{{t}^{2}}}{x} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      10. sub-neg78.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) + \left(-\frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

    13. Simplified78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}} \cdot t \]
  3. Recombined 4 regimes into one program.

  4. Final simplification80.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.000135:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-160} \lor \neg \left(t \leq 5.1 \cdot 10^{-18}\right):\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \end{array} \]

Alternative 7?

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -0.00082:\\ \;\;\;\;t \cdot \frac{-t_1}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-233} \lor \neg \left(t \leq 2.1 \cdot 10^{-159}\right) \land t \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if t < -8.1999999999999998e-4

    1. Initial program 34.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/34.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 19.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative42.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow242.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified42.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 91.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg91.4%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/91.4%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval91.4%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity91.4%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac91.4%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified91.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -8.1999999999999998e-4 < t < 3.60000000000000007e-233 or 2.0999999999999999e-159 < t < 5.8e-11

    1. Initial program 29.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/29.5%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified29.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 69.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+69.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow269.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out69.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow269.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow269.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/69.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg69.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow269.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative69.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow269.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef69.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified69.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 69.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/69.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-169.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow269.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    9. Simplified69.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. *-un-lft-identity69.0%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}}} \cdot t \]

      2. associate-/l*69.0%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      3. associate-/r/69.1%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      4. associate-/l*69.1%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(\color{blue}{\frac{t}{\frac{x}{t}}} + t \cdot t\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      5. +-commutative69.1%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      6. fma-def69.1%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right)} - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      7. associate-/r/69.1%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\ell \cdot \ell}{x}\right)}} \cdot t \]

      8. distribute-rgt-neg-in69.1%

        \[\leadsto \frac{\sqrt{2}}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{x}\right)}} \cdot t \]

    11. Applied egg-rr69.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]
    12. Step-by-step derivation

      1. *-lft-identity69.1%

        \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

      2. associate-+r-69.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}}} \cdot t \]

      3. associate-*l/69.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\color{blue}{\frac{\ell \cdot \ell}{x}} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      4. unpow269.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{\color{blue}{{\ell}^{2}}}{x} + 2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      5. fma-udef69.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      6. unpow269.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{{t}^{2}} + \frac{t}{x} \cdot t\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      7. +-commutative69.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \color{blue}{\left(\frac{t}{x} \cdot t + {t}^{2}\right)}\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      8. associate-*l/69.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\color{blue}{\frac{t \cdot t}{x}} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      9. unpow269.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{\color{blue}{{t}^{2}}}{x} + {t}^{2}\right)\right) - \frac{\ell \cdot \left(-\ell\right)}{x}}} \cdot t \]

      10. sub-neg69.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right) + \left(-\frac{\ell \cdot \left(-\ell\right)}{x}\right)}}} \cdot t \]

    13. Simplified69.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}} \cdot t \]

    if 3.60000000000000007e-233 < t < 2.0999999999999999e-159 or 5.8e-11 < t

    1. Initial program 25.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.4%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 18.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative33.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow233.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified33.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 88.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/88.1%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative88.1%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg88.1%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval88.1%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity88.1%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified88.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]
  3. Recombined 3 regimes into one program.

  4. Final simplification80.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00082:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-233} \lor \neg \left(t \leq 2.1 \cdot 10^{-159}\right) \land t \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

Alternative 8?

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := t \cdot \sqrt{\frac{2}{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t \cdot \frac{-t_3}{t}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-235}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + t_1}}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-155} \lor \neg \left(t \leq 5.4 \cdot 10^{-11}\right):\\ \;\;\;\;t \cdot \frac{t_3}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Derivation
  1. Split input into 5 regimes

  2. if t < -1.1000000000000001e-6

    1. Initial program 35.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/35.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 20.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow243.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified43.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 91.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg91.5%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/91.5%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative91.5%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg91.5%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval91.5%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity91.5%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac91.5%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified91.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -1.1000000000000001e-6 < t < -4.19999999999999977e-196 or 5.8999999999999999e-155 < t < 5.40000000000000009e-11

    1. Initial program 46.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/46.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified46.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
    4. Step-by-step derivation

      1. sqrt-undiv45.4%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \cdot t \]

      2. fma-udef45.4%

        \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

      3. +-commutative45.4%

        \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \cdot t \]

      4. *-commutative45.4%

        \[\leadsto \sqrt{\frac{2}{\color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1}} - \ell \cdot \ell}} \cdot t \]

      5. metadata-eval45.4%

        \[\leadsto \sqrt{\frac{2}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg45.4%

        \[\leadsto \sqrt{\frac{2}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      7. *-commutative45.4%

        \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \cdot t \]

    5. Applied egg-rr45.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

    6. Taylor expanded in x around inf 79.9%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    7. Step-by-step derivation

      1. distribute-lft-in79.9%

        \[\leadsto \sqrt{\frac{2}{\left(\frac{{\ell}^{2}}{x} + \color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]

      2. associate--l+79.9%

        \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      3. unpow279.9%

        \[\leadsto \sqrt{\frac{2}{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. +-commutative79.9%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow279.9%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. unpow279.9%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      7. associate-*r/79.9%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

    8. Simplified79.9%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    9. Taylor expanded in t around 0 79.7%

      \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. associate-*r/81.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-181.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow281.0%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    11. Simplified79.7%

      \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]

    if -4.19999999999999977e-196 < t < -1.15e-261

    1. Initial program 2.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/2.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified2.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 2.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*2.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative2.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg2.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval2.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative2.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow22.7%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified2.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
    7. Step-by-step derivation

      1. *-un-lft-identity2.7%

        \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

      2. sqrt-undiv2.7%

        \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

      3. +-commutative2.7%

        \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

    8. Applied egg-rr2.7%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
    9. Step-by-step derivation

      1. *-lft-identity2.7%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      2. associate-/r*2.7%

        \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      3. metadata-eval2.7%

        \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

      4. metadata-eval2.7%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

      5. sub-neg2.7%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

      6. unpow22.7%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

      7. associate-/r/2.7%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

      8. associate-*l/2.7%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

      9. *-commutative2.7%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

      10. unpow22.7%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

      11. sub-neg2.7%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      12. metadata-eval2.7%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

    10. Simplified2.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

    11. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2}}{t}} \cdot t \]
    12. Step-by-step derivation

      1. unpow20.0%

        \[\leadsto \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t} \cdot t \]

      2. rem-square-sqrt63.2%

        \[\leadsto \frac{\color{blue}{-1}}{t} \cdot t \]

    13. Simplified63.2%

      \[\leadsto \color{blue}{\frac{-1}{t}} \cdot t \]

    14. Taylor expanded in t around 0 63.4%

      \[\leadsto \color{blue}{-1} \]

    if -1.15e-261 < t < 4.9999999999999998e-235

    1. Initial program 2.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/2.0%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in x around inf 57.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate--l+57.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow257.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out57.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow257.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow257.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/57.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg57.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow257.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative57.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow257.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef57.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

    6. Simplified57.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    7. Taylor expanded in t around 0 56.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
    8. Step-by-step derivation

      1. sub-neg56.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(--1 \cdot \frac{{\ell}^{2}}{x}\right)}}} \cdot t \]

      2. unpow256.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(--1 \cdot \frac{{\ell}^{2}}{x}\right)}} \cdot t \]

      3. mul-1-neg56.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(-\color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      4. remove-double-neg56.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \color{blue}{\frac{{\ell}^{2}}{x}}}} \cdot t \]

      5. unpow256.8%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\color{blue}{\ell \cdot \ell}}{x}}} \cdot t \]

    9. Simplified56.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}} \cdot t \]

    if 4.9999999999999998e-235 < t < 5.8999999999999999e-155 or 5.40000000000000009e-11 < t

    1. Initial program 25.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.2%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 18.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow232.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified32.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 87.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/87.4%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative87.4%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg87.4%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval87.4%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity87.4%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified87.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]
  3. Recombined 5 regimes into one program.

  4. Final simplification81.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-196}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-235}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-155} \lor \neg \left(t \leq 5.4 \cdot 10^{-11}\right):\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \end{array} \]

Alternative 9?

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \frac{-t_2}{t}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-228} \lor \neg \left(t \leq 7.5 \cdot 10^{-155}\right) \land t \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_2}{t}\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if t < -7.40000000000000009e-7

    1. Initial program 35.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/35.7%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 20.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative43.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow243.1%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified43.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 91.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg91.5%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/91.5%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative91.5%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg91.5%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval91.5%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity91.5%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac91.5%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified91.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -7.40000000000000009e-7 < t < 2.6e-228 or 7.5000000000000006e-155 < t < 1.2000000000000001e-11

    1. Initial program 29.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/29.1%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified29.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
    4. Step-by-step derivation

      1. sqrt-undiv28.3%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \cdot t \]

      2. fma-udef28.3%

        \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

      3. +-commutative28.3%

        \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \cdot t \]

      4. *-commutative28.3%

        \[\leadsto \sqrt{\frac{2}{\color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1}} - \ell \cdot \ell}} \cdot t \]

      5. metadata-eval28.3%

        \[\leadsto \sqrt{\frac{2}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg28.3%

        \[\leadsto \sqrt{\frac{2}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      7. *-commutative28.3%

        \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \cdot t \]

    5. Applied egg-rr31.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

    6. Taylor expanded in x around inf 67.7%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
    7. Step-by-step derivation

      1. distribute-lft-in67.7%

        \[\leadsto \sqrt{\frac{2}{\left(\frac{{\ell}^{2}}{x} + \color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]

      2. associate--l+67.7%

        \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      3. unpow267.7%

        \[\leadsto \sqrt{\frac{2}{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. +-commutative67.7%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow267.7%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. unpow267.7%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      7. associate-*r/67.7%

        \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

    8. Simplified67.7%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

    9. Taylor expanded in t around 0 67.6%

      \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \color{blue}{-1 \cdot \frac{{\ell}^{2}}{x}}\right)}} \cdot t \]
    10. Step-by-step derivation

      1. associate-*r/68.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \cdot t \]

      2. neg-mul-168.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \cdot t \]

      3. unpow268.5%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \cdot t \]

    11. Simplified67.6%

      \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \cdot t \]

    if 2.6e-228 < t < 7.5000000000000006e-155 or 1.2000000000000001e-11 < t

    1. Initial program 25.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.2%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 18.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative32.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow232.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified32.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 87.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/87.4%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative87.4%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg87.4%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval87.4%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity87.4%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified87.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]
  3. Recombined 3 regimes into one program.

  4. Final simplification80.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-228} \lor \neg \left(t \leq 7.5 \cdot 10^{-155}\right) \land t \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

Alternative 10?

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{-t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.999999999999994e-310

    1. Initial program 33.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/33.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow230.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified30.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around -inf 74.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)\right)} \cdot t \]
    8. Step-by-step derivation

      1. mul-1-neg74.0%

        \[\leadsto \color{blue}{\left(-\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]

      2. associate-*r/74.0%

        \[\leadsto \left(-\color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}}\right) \cdot t \]

      3. +-commutative74.0%

        \[\leadsto \left(-\frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t}\right) \cdot t \]

      4. sub-neg74.0%

        \[\leadsto \left(-\frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      5. metadata-eval74.0%

        \[\leadsto \left(-\frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t}\right) \cdot t \]

      6. *-rgt-identity74.0%

        \[\leadsto \left(-\frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t}\right) \cdot t \]

      7. distribute-neg-frac74.0%

        \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    9. Simplified74.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]

    if -1.999999999999994e-310 < t

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 22.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow226.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified26.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 72.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/72.5%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative72.5%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg72.5%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval72.5%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity72.5%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified72.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]
  3. Recombined 2 regimes into one program.

  4. Final simplification73.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

Alternative 11?

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.999999999999994e-310

    1. Initial program 33.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/33.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow230.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified30.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
    7. Step-by-step derivation

      1. *-un-lft-identity30.2%

        \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

      2. sqrt-undiv30.1%

        \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

      3. +-commutative30.1%

        \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

    8. Applied egg-rr30.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
    9. Step-by-step derivation

      1. *-lft-identity30.1%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      2. associate-/r*30.1%

        \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      3. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

      4. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

      5. sub-neg30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

      6. unpow230.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

      7. associate-/r/39.1%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

      8. associate-*l/27.4%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

      9. *-commutative27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

      10. unpow227.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

      11. sub-neg27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      12. metadata-eval27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

    10. Simplified27.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

    11. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{t}\right)} \cdot t \]
    12. Step-by-step derivation

      1. unpow20.0%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t}\right) \cdot t \]

      2. rem-square-sqrt73.6%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{-1}}{t}\right) \cdot t \]

    13. Simplified73.6%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)} \cdot t \]

    if -1.999999999999994e-310 < t

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 22.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow226.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified26.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in t around 0 72.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \frac{1}{t}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate-*r/72.5%

        \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - 1}{1 + x}} \cdot 1}{t}} \cdot t \]

      2. +-commutative72.5%

        \[\leadsto \frac{\sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \cdot 1}{t} \cdot t \]

      3. sub-neg72.5%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \cdot 1}{t} \cdot t \]

      4. metadata-eval72.5%

        \[\leadsto \frac{\sqrt{\frac{x + \color{blue}{-1}}{x + 1}} \cdot 1}{t} \cdot t \]

      5. *-rgt-identity72.5%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{x + -1}{x + 1}}}}{t} \cdot t \]

    9. Simplified72.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x + -1}{x + 1}}}{t}} \cdot t \]
  3. Recombined 2 regimes into one program.

  4. Final simplification73.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

Alternative 12?

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{-1}{t \cdot x}\right)\right)\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.999999999999994e-310

    1. Initial program 33.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/33.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow230.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified30.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
    7. Step-by-step derivation

      1. *-un-lft-identity30.2%

        \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

      2. sqrt-undiv30.1%

        \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

      3. +-commutative30.1%

        \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

    8. Applied egg-rr30.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
    9. Step-by-step derivation

      1. *-lft-identity30.1%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      2. associate-/r*30.1%

        \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      3. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

      4. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

      5. sub-neg30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

      6. unpow230.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

      7. associate-/r/39.1%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

      8. associate-*l/27.4%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

      9. *-commutative27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

      10. unpow227.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

      11. sub-neg27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      12. metadata-eval27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

    10. Simplified27.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

    11. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{t}\right)} \cdot t \]
    12. Step-by-step derivation

      1. unpow20.0%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t}\right) \cdot t \]

      2. rem-square-sqrt73.6%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{-1}}{t}\right) \cdot t \]

    13. Simplified73.6%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)} \cdot t \]

    if -1.999999999999994e-310 < t

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 22.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow226.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified26.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in x around inf 72.1%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{t \cdot {x}^{2}} + \frac{1}{t}\right) - \frac{1}{t \cdot x}\right)} \cdot t \]
    8. Step-by-step derivation

      1. associate--l+72.1%

        \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{t \cdot {x}^{2}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right)} \cdot t \]

      2. associate-*r/72.1%

        \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{t \cdot {x}^{2}}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      3. metadata-eval72.1%

        \[\leadsto \left(\frac{\color{blue}{0.5}}{t \cdot {x}^{2}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      4. unpow272.1%

        \[\leadsto \left(\frac{0.5}{t \cdot \color{blue}{\left(x \cdot x\right)}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

    9. Simplified72.1%

      \[\leadsto \color{blue}{\left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right)} \cdot t \]
  3. Recombined 2 regimes into one program.

  4. Final simplification72.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{-1}{t \cdot x}\right)\right)\\ \end{array} \]

Alternative 13?

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.999999999999994e-310

    1. Initial program 33.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/33.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow230.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified30.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
    7. Step-by-step derivation

      1. *-un-lft-identity30.2%

        \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

      2. sqrt-undiv30.1%

        \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

      3. +-commutative30.1%

        \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

    8. Applied egg-rr30.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
    9. Step-by-step derivation

      1. *-lft-identity30.1%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      2. associate-/r*30.1%

        \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      3. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

      4. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

      5. sub-neg30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

      6. unpow230.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

      7. associate-/r/39.1%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

      8. associate-*l/27.4%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

      9. *-commutative27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

      10. unpow227.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

      11. sub-neg27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      12. metadata-eval27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

    10. Simplified27.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

    11. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{t}\right)} \cdot t \]
    12. Step-by-step derivation

      1. unpow20.0%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t}\right) \cdot t \]

      2. rem-square-sqrt73.6%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{-1}}{t}\right) \cdot t \]

    13. Simplified73.6%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)} \cdot t \]

    if -1.999999999999994e-310 < t

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 22.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow226.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified26.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in x around inf 70.9%

      \[\leadsto \color{blue}{\frac{1}{t}} \cdot t \]

    8. Taylor expanded in t around 0 71.0%

      \[\leadsto \color{blue}{1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification72.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14?

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \frac{-1}{t \cdot x}\right)\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.999999999999994e-310

    1. Initial program 33.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/33.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow230.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified30.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
    7. Step-by-step derivation

      1. *-un-lft-identity30.2%

        \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

      2. sqrt-undiv30.1%

        \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

      3. +-commutative30.1%

        \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

    8. Applied egg-rr30.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
    9. Step-by-step derivation

      1. *-lft-identity30.1%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      2. associate-/r*30.1%

        \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      3. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

      4. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

      5. sub-neg30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

      6. unpow230.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

      7. associate-/r/39.1%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

      8. associate-*l/27.4%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

      9. *-commutative27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

      10. unpow227.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

      11. sub-neg27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      12. metadata-eval27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

    10. Simplified27.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

    11. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{t}\right)} \cdot t \]
    12. Step-by-step derivation

      1. unpow20.0%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t}\right) \cdot t \]

      2. rem-square-sqrt73.6%

        \[\leadsto \left(\frac{1}{t \cdot x} + \frac{\color{blue}{-1}}{t}\right) \cdot t \]

    13. Simplified73.6%

      \[\leadsto \color{blue}{\left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)} \cdot t \]

    if -1.999999999999994e-310 < t

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 22.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow226.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified26.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in x around inf 71.6%

      \[\leadsto \color{blue}{\left(\frac{1}{t} - \frac{1}{t \cdot x}\right)} \cdot t \]
  3. Recombined 2 regimes into one program.

  4. Final simplification72.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(\frac{1}{t \cdot x} + \frac{-1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \frac{-1}{t \cdot x}\right)\\ \end{array} \]

Alternative 15?

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.999999999999994e-310

    1. Initial program 33.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/33.3%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative30.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow230.2%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified30.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
    7. Step-by-step derivation

      1. *-un-lft-identity30.2%

        \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

      2. sqrt-undiv30.1%

        \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

      3. +-commutative30.1%

        \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

    8. Applied egg-rr30.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
    9. Step-by-step derivation

      1. *-lft-identity30.1%

        \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      2. associate-/r*30.1%

        \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

      3. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

      4. metadata-eval30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

      5. sub-neg30.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

      6. unpow230.1%

        \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

      7. associate-/r/39.1%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

      8. associate-*l/27.4%

        \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

      9. *-commutative27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

      10. unpow227.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

      11. sub-neg27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      12. metadata-eval27.4%

        \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

    10. Simplified27.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

    11. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2}}{t}} \cdot t \]
    12. Step-by-step derivation

      1. unpow20.0%

        \[\leadsto \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t} \cdot t \]

      2. rem-square-sqrt73.2%

        \[\leadsto \frac{\color{blue}{-1}}{t} \cdot t \]

    13. Simplified73.2%

      \[\leadsto \color{blue}{\frac{-1}{t}} \cdot t \]

    14. Taylor expanded in t around 0 73.3%

      \[\leadsto \color{blue}{-1} \]

    if -1.999999999999994e-310 < t

    1. Initial program 25.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
    2. Step-by-step derivation

      1. associate-*l/25.6%

        \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

    3. Simplified25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

    4. Taylor expanded in t around inf 22.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
    5. Step-by-step derivation

      1. associate-/l*26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative26.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow226.6%

        \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

    6. Simplified26.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

    7. Taylor expanded in x around inf 70.9%

      \[\leadsto \color{blue}{\frac{1}{t}} \cdot t \]

    8. Taylor expanded in t around 0 71.0%

      \[\leadsto \color{blue}{1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification72.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16?

\[-1 \]
Derivation

  1. Initial program 29.5%

    \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
  2. Step-by-step derivation

    1. associate-*l/29.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

  3. Simplified29.6%

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

  4. Taylor expanded in t around inf 25.4%

    \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
  5. Step-by-step derivation

    1. associate-/l*28.4%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

    2. +-commutative28.4%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

    3. sub-neg28.4%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

    4. metadata-eval28.4%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

    5. +-commutative28.4%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

    6. unpow228.4%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

  6. Simplified28.4%

    \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]
  7. Step-by-step derivation

    1. *-un-lft-identity28.4%

      \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right)} \cdot t \]

    2. sqrt-undiv28.4%

      \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}}\right) \cdot t \]

    3. +-commutative28.4%

      \[\leadsto \left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + -1}}{t \cdot t}}}}\right) \cdot t \]

  8. Applied egg-rr28.4%

    \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\right)} \cdot t \]
  9. Step-by-step derivation

    1. *-lft-identity28.4%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

    2. associate-/r*28.4%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{2}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}}} \cdot t \]

    3. metadata-eval28.4%

      \[\leadsto \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{\frac{x + -1}{t \cdot t}}}} \cdot t \]

    4. metadata-eval28.4%

      \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x + \color{blue}{\left(-1\right)}}{t \cdot t}}}} \cdot t \]

    5. sub-neg28.4%

      \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{\color{blue}{x - 1}}{t \cdot t}}}} \cdot t \]

    6. unpow228.4%

      \[\leadsto \sqrt{\frac{1}{\frac{x + 1}{\frac{x - 1}{\color{blue}{{t}^{2}}}}}} \cdot t \]

    7. associate-/r/35.8%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{x + 1}{x - 1} \cdot {t}^{2}}}} \cdot t \]

    8. associate-*l/24.7%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]

    9. *-commutative24.7%

      \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{{t}^{2} \cdot \left(x + 1\right)}}{x - 1}}} \cdot t \]

    10. unpow224.7%

      \[\leadsto \sqrt{\frac{1}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(x + 1\right)}{x - 1}}} \cdot t \]

    11. sub-neg24.7%

      \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

    12. metadata-eval24.7%

      \[\leadsto \sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \cdot t \]

  10. Simplified24.7%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + -1}}}} \cdot t \]

  11. Taylor expanded in x around -inf 0.0%

    \[\leadsto \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2}}{t}} \cdot t \]
  12. Step-by-step derivation

    1. unpow20.0%

      \[\leadsto \frac{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}{t} \cdot t \]

    2. rem-square-sqrt38.6%

      \[\leadsto \frac{\color{blue}{-1}}{t} \cdot t \]

  13. Simplified38.6%

    \[\leadsto \color{blue}{\frac{-1}{t}} \cdot t \]

  14. Taylor expanded in t around 0 38.6%

    \[\leadsto \color{blue}{-1} \]

  15. Final simplification38.6%

    \[\leadsto -1 \]

Reproduce

?
herbie shell --seed 2023166 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))